Feed forward parameter values for use in theoretically generating spectra

ABSTRACT

A method of controlling a polishing operation is described. A controller stores an optical model for a layer stack having a plurality of layers and a plurality of input parameters including a first parameter and a second parameter. The controller stores data defining a plurality of default values for the first parameter and measures an optical property of a substrate and generates a second value. Using the optical model and the second value and iterating over the first values, a number of reference spectra are calculated. A spectrum is measured and the measured spectrum is matched to the reference spectra and the best matched reference spectrum is determined. The first value of the best matched reference spectrum is determined and is used to adjust a polishing endpoint or a polishing parameter of a polishing apparatus.

TECHNICAL FIELD

The present disclosure relates to optical monitoring, e.g., for control of chemical mechanical polishing of substrates.

BACKGROUND

An integrated circuit is typically formed on a substrate by the sequential deposition of conductive, semiconductive, or insulative layers on a silicon wafer. One fabrication step involves depositing a filler layer over a non-planar surface and planarizing the filler layer. For certain applications, the filler layer is planarized until the top surface of a patterned layer is exposed. A conductive filler layer, for example, can be deposited on a patterned insulative layer to fill the trenches or holes in the insulative layer. After planarization, the portions of the conductive layer remaining between the raised pattern of the insulative layer form vias, plugs, and lines that provide conductive paths between thin film circuits on the substrate. For other applications, such as oxide polishing, the filler layer is planarized until a predetermined thickness is left over the non-planar surface. In addition, planarization of the substrate surface is usually required for photolithography.

Chemical mechanical polishing (CMP) is one accepted method of planarization. This planarization method typically requires that the substrate be mounted on a carrier head. The exposed surface of the substrate is typically placed against a rotating polishing pad. The carrier head provides a controllable load on the substrate to push it against the polishing pad. A polishing liquid, such as a slurry with abrasive particles, is typically supplied to the surface of the polishing pad.

One problem in CMP is determining whether the polishing process is complete, i.e., whether a substrate layer has been planarized to a desired flatness or thickness, or when a desired amount of material has been removed. Variations in the initial thickness of the substrate layer, the slurry composition, the polishing pad condition, the relative speed between the polishing pad and the substrate, and the load on the substrate can cause variations in the material removal rate. These variations cause variations in the time needed endpoint merely as a function of polishing time.

In some systems, a substrate is optically monitored in-situ during polishing, e.g., through a window in the polishing pad. However, existing optical monitoring techniques may not satisfy increasing demands of semiconductor device manufacturers.

SUMMARY

In some optical monitoring processes, a measured spectrum is compared to a library of reference spectra to find the best matching reference spectrum. One technique to build a library of reference spectra is to calculate a reference spectrum based on an optical model of the thin film stack on the substrate. The size of the library of reference spectra grows rapidly as the number of layers, the number of variable parameters for each layer, or the number of increments each variable spans increase. The larger the library, the more memory needed to hold the library and the higher the processing load to search the library. The larger library may also produce multiple possible matches, causing lower reliability or requiring other techniques to sift the possible matches and select an acceptable match.

Another optical monitoring technique is to fit a function, e.g., the optical model, to the measured spectra. However, a complex optical model with a large the number of variable input parameters can suffer similar problems of computational load and possible false fitting of the optical parameters.

In one aspect a method of controlling a polishing operation includes storing an optical model for a layer stack having a plurality of layers, the optical model having a plurality of input parameters including a first parameter and a second parameter, storing data defining a plurality of default first values for the first parameter, measuring an optical property of a substrate at a stand-alone or in-sequence metrology station to generate a second value for the second parameter, for each first value from the plurality of first values, calculating a reference spectrum using the optical model based on the first value and the second value, to generate a plurality of reference spectra, transporting the substrate from the stand-alone or in-sequence metrology station to an in-sequence or in-situ monitoring system of a chemical mechanical polishing apparatus, measuring a spectrum with the in-sequence or in-situ monitoring system to provide a measured spectrum, determining a best matching reference spectrum from the plurality of reference spectra that provides a best match to the measured spectrum, determining the first value associated with the best matching reference spectrum, polishing the substrate with the polishing apparatus, and adjusting a polishing endpoint or a polishing parameter of the polishing apparatus based on the first value associated with the best matching reference spectrum.

Implementations may include one or more of the following features. The first parameter includes a thickness of an outermost layer of the substrate and the second parameter includes an index of refraction or extinction coefficient of the outermost layer or a thickness of an underlying layer of the substrate. The optical property of the substrate is measured at the stand-alone metrology station to generate the second value, and the spectrum is measured with the in-sequence monitoring system to provide the measured spectrum. Alternatively, the optical property of the substrate is measured at the in-sequence metrology station to generate the second value, and the spectrum is measured with the in-situ monitoring system to provide the measured spectrum. A plurality of second values are calculated based on the second value, a default range, and a default increment. A reference spectrum is calculated using the optical model for each combination of a first value from the plurality of first values and a second value from the plurality of second values.

In another aspect a method of controlling a polishing operation includes storing an optical model for a layer stack having a plurality of layers, the optical model having a plurality of input parameters including a first parameter and a second parameter, measuring an optical property of a substrate at a stand-alone or in-sequence metrology station to generate a second value for a second parameter of the plurality of optical parameters, transporting the substrate from the stand-alone or in-sequence metrology station to an in-sequence or in-situ monitoring system of a chemical mechanical polishing apparatus, measuring a spectrum with the in-sequence or in-situ monitoring system to provide a measured spectrum, fitting the optical model to the measured spectrum, the fitting including finding a first value of the first parameter that provides a minimum difference between an output spectrum of the optical model and the measured spectrum, the fitting including holding the second parameter at the second value or using the second value of the second parameter as a seed value in searching for the minimum difference, polishing the substrate with the polishing apparatus, and adjusting a polishing endpoint or a polishing parameter of the polishing apparatus based on the first value associated with the fitted optical model.

Implementations may include one or more of the following features. The first parameter includes a thickness of an outermost layer of the substrate and the second parameter includes an index of refraction or extinction coefficient of the outermost layer or a thickness of an underlying layer of the substrate. The optical property of the substrate is measured at the stand-alone metrology station to generate the second value, and the spectrum is measured with the in-sequence monitoring system to provide the measured spectrum. Alternatively, the optical property of the substrate is measured at the in-sequence metrology station to generate the second value, and the spectrum is measured with the in-situ monitoring system to provide the measured spectrum. The fitting includes holding the second parameter at the second value. Alternatively, the fitting includes using the second value of the second parameter as a seed value in searching for the minimum difference.

Certain implementations can include one or more of the following advantages. A library of reference spectra that spans the likely range of variations of incoming substrates can be calculated from an optical model. The size of the library may be reduced if a variable parameter is replaced by a known value. The variable can be determined from an upstream measurement before polishing or measurement at an in-line monitoring station. Thus, reliability of the endpoint system to detect a desired polishing endpoint may be improved, and within-wafer and wafer-to-wafer thickness non-uniformity (WIWNU and WTWNU) may be reduced. Additionally, the information can be used for in-sequence monitoring to determine the polish time/pressures for the next platen or for a rework. In addition, processing load for calculation of the layer stacks can be reduced. Additionally, prediction accuracy can increase as a set of parameters are predetermined which reduces the chance of incorrect fits or correlations.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a schematic cross-sectional view of an example of a polishing apparatus.

FIG. 2 illustrates a schematic cross-sectional view of an example of an in-sequence optical metrology station.

FIG. 3 illustrates a schematic top view of a substrate having multiple zones.

FIG. 4 illustrates a measured spectrum from the in-situ optical monitoring system.

FIG. 5 illustrates a library of reference spectra.

FIG. 6 illustrates an index trace.

FIG. 7 illustrates an index trace having a linear function fit to index values collected after clearance of an overlying layer is detected.

FIG. 8 is a flow graph of a method for controlling a polishing operation of a product substrate.

FIG. 9 is a flow graph of a method for controlling a polishing operation.

Like reference numbers and designations in the various drawings indicate like elements.

DETAILED DESCRIPTION

One optical monitoring technique for controlling a polishing operation is to measure a spectrum of light reflected from a substrate, either in-situ during polishing or at an in-line metrology station, and compare the measured spectrum to a plurality of reference spectra from a library, and identify a best-matching reference spectrum. The reference spectra can be calculated from an optical model by varying multiple input parameters. Ideally, the optical model has sufficient degrees of freedom to generate a library of reference spectra that spans the likely range of variation of incoming substrates.

One potential problem is the size of the library of reference spectra. For reliable polishing of some types of substrates, the library of reference spectra may have reference spectra that span the likely range of variations in thickness and/or other parameters of the underlying layers of incoming substrates, and consequently increasing the size of the library. In some examples, for a reliable match, the numbers of increments of the variable parameters increase and consequently the size of the library of reference spectra further increases. The increase in library size may hamper a reliable real-time match if all library members are considered.

Another optical monitoring technique is to fit a function, e.g., an optical model, to the measured spectra. If the optical model is complex and has a large number of parameters to optimize, the function fitting technique can have a related problem; computational load and the possibility of false matches.

It may be possible to reduce the size of the library or simplify the fitting of the optical model to the measured spectrum if a variable is replaced with a measured value. Additionally, prediction accuracy can increase as a set of parameters are predetermined which reduces the chance of incorrect fits or correlations.

Some substrates include regions with different stacks of layers. Examples of materials for layers from the stack include silicon oxide, carbon-doped silicon oxide, silicon carbide, silicon nitride, carbon-doped silicon nitride, polysilicon, and metal, e.g., copper. As a very simple example, some regions can include a single dielectric layer over a metal layer, and other regions can include two dielectric layers over a metal layer. Of course, much more complex layer stacks are likely in a real-world application. For example, when polishing a substrate in a back-end-of-line process, some regions of the substrate can include exposed metal, other regions can include a single layer set, and yet other regions can include multiple vertically arranged layer sets. Each layer set can correspond to a metal layer in the metal interconnect structure of the substrate. For example, each layer set includes a dielectric layer, e.g., a low-k dielectric, and an etch-stop layer, e.g., silicon carbide, silicon nitride, or carbon-silicon nitride (SiCN).

FIG. 1 illustrates an example of a polishing apparatus 100. The polishing apparatus 100 includes a rotatable disk-shaped platen 120 on which a polishing pad 110 is situated. The platen is operable to rotate about an axis 125. For example, a motor 121 can turn a drive shaft 124 to rotate the platen 120. The polishing pad 110 can be a two-layer polishing pad with an outer polishing layer 112 and a softer backing layer 114.

The polishing apparatus 100 can include a port 130 to dispense polishing liquid 132, such as a slurry, onto the polishing pad 110 to the pad. The polishing apparatus can also include a polishing pad conditioner to abrade the polishing pad 110 to maintain the polishing pad 110 in a consistent abrasive state.

The polishing apparatus 100 includes one or more carrier heads 140. Each carrier head 140 is operable to hold a substrate 10 against the polishing pad 110. Each carrier head 140 can have independent control of the polishing parameters, for example pressure, associated with each respective substrate.

In particular, each carrier head 140 can include a retaining ring 142 to retain the substrate 10 below a flexible membrane 144. Each carrier head 140 also includes a plurality of independently controllable pressurizable chambers defined by the membrane, e.g., three chambers 146 a-146 c, which can apply independently controllable pressurizes to associated zones 148 a-148 c on the flexible membrane 144 and thus on the substrate 10 (see FIG. 3). Referring to FIG. 3, the center zone 148 a can be substantially circular, and the remaining zones 148 b-148 c can be concentric annular zones around the center zone 148 a. Although only three chambers are illustrated in FIGS. 1 and 3 for ease of illustration, there could be one or two chambers, or four or more chambers, e.g., five chambers.

Returning to FIG. 1, each carrier head 140 is suspended from a support structure 150, e.g., a carousel or track, and is connected by a drive shaft 152 to a carrier head rotation motor 154 so that the carrier head can rotate about an axis 155. Optionally each carrier head 140 can oscillate laterally, e.g., on sliders on the carousel 150; by rotational oscillation of the carousel itself, or by motion of a carriage 108 (see FIG. 2) that supports the carrier head 140 along the track. In operation, the platen is rotated about its central axis 125, and each carrier head is rotated about its central axis 155 and translated laterally across the top surface of the polishing pad.

In some implementations, the polishing apparatus includes an in-situ optical monitoring system 160, e.g., a spectrographic monitoring system, which can be used to measure a spectrum of reflected light from a substrate undergoing polishing. An optical access through the polishing pad is provided by including an aperture (i.e., a hole that runs through the pad) or a solid window 118.

If the window 118 is installed in the platen, due to the rotation of the platen (shown by arrow 204), as the window 108 travels below a carrier head, the optical monitoring system making spectra measurements at a sampling frequency will cause the spectra measurements to be taken at locations 201 in an arc that traverses the substrate 10.

In some implementation, illustrated in FIG. 2, the polishing apparatus includes an in-sequence optical monitoring system 160 having a probe 180 positioned between two polishing stations or between a polishing station and a transfer station. The probe 180 of the in-sequence monitoring system 160 can be supported on a platform 106, and can be positioned on the path of the carrier head.

The probe 180 can include a mechanism to adjust its vertical height relative to the top surface of the platform 106. In some implementations, the probe 180 is supported on an actuator system 182 that is configured to move the probe 180 laterally in a plane parallel to the plane of the track 128. The actuator system 182 can be an XY actuator system that includes two independent linear actuators to move probe 180 independently along two orthogonal axes. In some implementations, there is no actuator system 182, and the probe 180 remains stationary (relative to the platform 106) while the carrier head 126 moves to cause the spot measured by the probe 180 to traverse a path on the substrate.

Referring to FIGS. 1 and 2, in either of the in-situ or in-sequence implementations, the optical monitoring system 160 can include a light source 162, a light detector 164, and circuitry 166 for sending and receiving signals between a remote controller 190, e.g., a computer, and the light source 162 and light detector 164. One or more optical fibers can be used to transmit the light from the light source 162 to the optical access in the polishing pad, and to transmit light reflected from the substrate 10 to the detector 164. For example, a bifurcated optical fiber 170 can be used to transmit the light from the light source 162 to the substrate 10 and back to the detector 164. The bifurcated optical fiber an include a trunk 172 positioned in proximity to the optical access, and two branches 174 and 176 connected to the light source 162 and detector 164, respectively. The probe 180 can include the trunk end of the bifurcated optical fiber.

The light source 162 can be operable to emit white light. In one implementation, the white light emitted includes light having wavelengths of 200-800 nanometers. In some implementations, the light source 162 generates unpolarized light. In some implementations, a polarization filter 178 (illustrated in FIG. 2, although it can be used in the monitoring system of FIG. 1) can be positioned between the light source 162 and the substrate 10. A suitable light source is a xenon lamp or a xenon mercury lamp.

The light detector 164 can be a spectrometer. A spectrometer is an optical instrument for measuring intensity of light over a portion of the electromagnetic spectrum. A suitable spectrometer is a grating spectrometer. Typical output for a spectrometer is the intensity of the light as a function of wavelength (or frequency). FIG. 4 illustrates an example of a measured spectrum 300.

As noted above, the light source 162 and light detector 164 can be connected to a computing device, e.g., the controller 190, operable to control their operation and receive their signals. The computing device can include a microprocessor situated near the polishing apparatus, e.g., a programmable computer. In operation, the controller 190 can receive, for example, a signal that carries information describing a spectrum of the light received by the light detector for a particular flash of the light source or time frame of the detector.

Optionally, the in-sequence metrology system 160 can be a wet metrology system. In a wet-metrology system, measurement of the surface of the substrate is conducted while a layer of liquid covers the portion of the surface being measured. An advantage of wet metrology is that the liquid can have a similar index of refraction as the optical fiber 270. The liquid can provide a homogeneous medium through which light can travel to and from the surface of the film that is to be or that has been polished. The wet metrology system can be configured such that the liquid is flowing during the measurement. A flowing liquid can flush away polishing residue, e.g., slurry, from the surface of the substrate being measured.

In some implementations, the controller 190, e.g., the computing device, can be programmed to compare a measured spectrum to multiple reference spectra and to determine which reference spectrum provides the best match.

In some implementations, controller software can be used to automatically calculate multiple reference spectra. Since there are variations in the thicknesses of the underlying layers of the incoming substrates, the manufacturer can input a thickness range and a thickness increment or a number of increments for at least one of the underlying layers, e.g., for multiple underlying layers. The software will calculate a reference spectra for each combination of thicknesses of the underlying layers. Multiple reference spectra can be calculated for each thickness of the overlying layer.

To calculate the reference spectra, the following optical model can be used. The reflectance R_(STACK) of the top layer p of a thin film stack can be calculated as

$R_{STACK} = {\frac{E_{p}^{-}}{E_{p}^{+}}}^{2}$

where Ep+ represents the electro-magnetic field strength of the incoming light beam and Ep− represents the electromagnetic field strength of the outgoing light beam.

The values Ep⁺ and E_(p) ⁻ can be calculated as

E _(p) ⁺=(E _(p) +H _(p)/μ_(p))/2 E _(p) ⁻=(E _(p) −H _(p)/μ_(p))/2

The fields E and H in an arbitrary layer j can be calculated using transfer-matrix methods from the fields E and H in an underlying layer. Thus, in a stack of layers 0, 1, . . . , p−1, p (where layer 0 is the bottom layer and layer p is the outermost layer), for a given layer j>0, E_(j) and H_(j) can be calculated as

$\begin{bmatrix} E_{j} \\ H_{j} \end{bmatrix} = {\begin{bmatrix} {\cos \; g_{j}} & {\frac{}{u_{j}}\sin \; g_{j}} \\ {{\mu}_{j}\sin \; g_{j}} & {\cos \; g_{j}} \end{bmatrix}\begin{bmatrix} E_{j - 1} \\ H_{j - 1} \end{bmatrix}}$

with μ_(j)=(n_(j)−ik_(j))·cos φ_(j) and g_(j)=2π(n_(j)−ik_(j))·t_(j)·cos φ_(j)/λ, where n_(j) is the index of refraction of layer j, k_(J) is an extinction coefficient of layer j, t_(j) is the thickness of layer j, φ_(j) is the incidence angle of the light to layer j, and λ is the wavelength. For the bottom layer in the stack, i.e., layer j=0, E₀=1 and H₀=μ₀=(n₀−ik₀)·cos φ₀. The index of refraction n and the extinction coefficient k for each layer can be determined from scientific literature, and can be functions of wavelength. The incidence angle φ can be calculated from Snell's law.

The thickness t for a layer can be calculated from the thickness range and thickness increment input by the user for the layer, e.g., tj=TMINj+k*TINCj for k=0, 1, . . . , for tj≦TMAXj, where TMINj and TMAXj are the lower and upper boundaries of the range of thicknesses for layer j and TINCj is the thickness increment for layer j. The calculation can be iterated for each combination of thickness values of the layers.

A potential advantage of this technique is quick generation of a large number of reference spectra that can correspond to different combinations of thicknesses of layers on the substrate, thus improving likelihood of finding a good matching reference spectra and improving accuracy and reliability of the optical monitoring system.

As an example, the light intensity reflected from the substrate can be calculated as

$\begin{bmatrix} E_{5} \\ H_{5} \end{bmatrix} = {{{\begin{bmatrix} {\cos \; g_{4}} & {\frac{}{u_{j}}\sin \; g_{4}} \\ {{\mu}_{4}\sin \; g_{4}} & {\cos \; g_{4}} \end{bmatrix}\begin{bmatrix} {\cos \; g_{3}} & {\frac{}{u_{j}}\sin \; g_{3}} \\ {{\mu}_{3}\sin \; g_{3}} & {\cos \; g_{3}} \end{bmatrix}}\begin{bmatrix} {\cos \; g_{2}} & {\frac{}{u_{j}}\sin \; g_{2}} \\ {{\mu}_{2}\sin \; g_{2}} & {\cos \; g_{2}} \end{bmatrix}}{\quad{\begin{bmatrix} {\cos \; g_{1}} & {\frac{}{u_{j}}\sin \; g_{1}} \\ {{\mu}_{1}\sin \; g_{1}} & {\cos \; g_{1}} \end{bmatrix}\begin{bmatrix} 1 \\ \mu_{0} \end{bmatrix}}}}$

with values of g₄ and μ₄ depending on the thickness, index of refraction and extinction coefficient of the outermost layer of the substrate, e.g., an upper dielectric layer, e.g., a low-k material, g₃ and μt₃ depending on the thickness, index of refraction and extinction coefficient of an underlying layer, e.g., an etch stop layer, e.g., SiCN, g₂ and μ₂ depending on the thickness, index of refraction and extinction coefficient of another underlying layer, e.g., a lower dielectric layer, g₁ and μ₁ depending on the thickness, index of refraction and extinction coefficient of another underlying layer, e.g., a passiviation layer, e.g., SiN, and μ₀ depending on the index of refraction and extinction coefficient of the bottom layer, e.g., a conductive layer, e.g., copper.

The reflectance R_(STACK) can then be calculated as

$R_{STACK} = \frac{E_{5} - \frac{H_{5}}{\mu_{5}}}{E_{5} + \frac{H_{5}}{\mu_{5}}}$

Although not shown, the presence of a layer of water over the substrate (to represent the polishing liquid through which the light will be arriving) can also be accounted for in the optical model, e.g., during in-situ monitoring.

The substrate and associated optical stack described above is only one possible assembly of layers, and many others are possible. For example, the optical stack described above uses a conductive layer at the bottom of the optical stack, which would be typical for a substrate in a back-end-of-line process. However, in a front-end-of-line process, or if the conductive layer is a transparent material, then the bottom of the optical stack can be the semiconductor wafer, e.g., silicon. As another example, some substrates may not include the lower dielectric layer.

In some situations, some spectral measurements may be made from substrates with a layer having a higher index of refraction or extinction coefficient, whereas other spectral calculations may be made from substrates with a layer having a lower index of refraction or extinction coefficient. Therefore, in addition to variations of the layer thicknesses, the above described optical model can include variations in the index of refraction n and/or the extinction coefficient k of one or more layers in the optical stack. The one or more layers can include the underlying layer and/or the overlying layer.

The controller software may receive user input identifying a first number of different contribution percentages for the first stack, and a plurality of different contribution percentages for the second stack can be calculated from the first number of different contribution percentages.

In some implementations, controller software can be used to receive user input to identify a set of one or more refractive index functions and/or a set of one or more extinction coefficient functions. A refractive index function can provide a refractive index for a material of a layer as a function of wavelength. Similarly, an extinction coefficient function can provide an extinction coefficient for a material of a layer as a function of wavelength. Where there are variations between substrates in the refractive index, a plurality of different refractive index functions can be used to generate the reference spectra. Similarly, where there are variations between substrates in the extinction coefficient, a plurality of extinction coefficient functions can be used to generate the reference spectra. For example, the controller software can calculate a reference spectrum for each combination of a refractive index function from the set of refractive index functions and an extinction coefficient function from the set of extinction coefficient functions.

The different refractive index functions can be variants of a common generic refractive index function. For example, the generic refractive index function can be a function of wavelength and one or more additional coefficients, and the different refractive index functions can constitute different values for the coefficient(s). The values of the coefficient(s) can be set by a user, e.g., the semiconductor manufacturer. For example, for a particular coefficient, a user can set the values by inputting a lower value, an upper value and an incremental value or a number of total increments. In another example a user can define an index range and an extinction coefficient range. The user can also define an index increment and an extinction coefficient increment or alternatively can define a number of increments for each one of index and extinction coefficient ranges.

In addition to variations of the layer thicknesses and the layer indices of refraction and/or the extinction coefficients, the optical model can include variations in the spectral contribution of the metal layer. That is, depending on the pattern on the die being manufactured, some spectral measurements may be made in regions with high concentration of metal (e.g., from metal material in the trenches), whereas other spectral measurements may be made in regions with lower concentration of metal.

During monitoring process, the placement of the light beam on the substrate is not precisely controlled. Consequently the light beam will sometimes land primary on a region with one layer stack, and sometimes the light beam will land primarily on a region with a different layer stack. In short, the percentage contribution to the spectrum from each different layer stack on the substrate can vary from measurement to measurement. However, it is possible to generate multiple reference spectra that span the likely range of variation in contribution by the different layer stacks.

In some implementations, the optical model can include variations in the spectral contribution of the different layer stacks. That is, depending on the pattern on the die being manufactured, some spectral measurements may be made in regions with high percentage (by area) of a first layer stack, whereas other spectral measurements may be made in regions with lower concentration of the first layer stack.

The spectrum R_(LIBRARY) that is added to the library can be a combination of multiple stack models. For example, there could be a first layer stack, R_(STACK1) which is the spectral contribution of the topmost layer set, and a second layer stack R_(STACK2) which is the spectral contribution of the two topmost layer sets. For example, the first layer set can include a capping layer, a dielectric layer, and a barrier layer (and copper as the bottom of the stack). The second layer set can include the capping layer, dielectric layer and barrier layer from the first stack, plus the dielectric layer and barrier layer that would reside beneath the first stack (and again, copper as the bottom of the stack).

The spectrum R_(LIBRARY) that is added to the library ban be calculated as

$R_{LIBRARY} = {\frac{1}{R_{REFERENCE}}\left\lbrack {{X*R_{{STACK}\; 1}} + {\left( {1 - X} \right)*R_{{STACK}\; 2}}} \right\rbrack}$

where R_(STACK1) is the first spectrum, R_(STACK2) is the second spectrum, R_(REFERENCE) is a spectrum of a bottom layer of the first stack and the second stack, and X is the percentage contribution for the first stack. The calculation of spectrum R_(LIBRARY) can be iterated over multiple values for X. For example, X can vary between 0.0 and 1.0 at 0.1 intervals. The controller software may receive user input identifying a first number of different contribution percentages for the first stack, and a plurality of different contribution percentages for the second stack can be calculated from the first number of different contribution percentages.

A potential advantage of this technique is generation of reference spectra that can correspond to different percentage contributions by different layer stacks in the measured spot on the substrate, thus improving likelihood of finding a good matching reference spectra and improving accuracy and reliability of the optical monitoring system.

In addition to variations of the layer thicknesses, the optical model can include variations in the spectral contribution of the metal layer. That is, depending on the pattern on the die being manufactured, some spectral measurements may be made in regions with high concentration of metal (e.g., from metal material in the trenches), whereas other spectral measurements may be made in regions with lower concentration of metal. As a layer of material is defined by refractive index, the extinction coefficient and thickness, for a given material there is a function for each one of refractive index and extinction coefficient that characterize its optical properties. The functions can either be measured, empirically determined, or modeled.

So the calculation for R_(LIBRARY) could look something like:

$R_{LIBRARY} = {\frac{1}{R_{REFERENCE}}\left\lbrack {{Y*R_{METAL}} + {X*R_{{STACK}\; 1}} + {\left( {1 - X - Y} \right)*R_{{STACK}\; 2}}} \right\rbrack}$

where X+Y<1, R_(STACK1) is the first spectrum, R_(STACK2) is the second spectrum, R_(METAL) is the third spectrum, R_(REFERENCE) is a spectrum of a bottom layer of the stack, and X is the percentage contribution for the first stack, and Y is the percentage contribution for the metal. In some implementations, e.g., if the metal layers of adjacent regions are the same material, e.g., copper, then R_(REFERENCE) and R_(METAL) are the same spectrum, e.g., the spectrum for copper.

In some implementations, calculation of the second spectrum can ignore layers below the second layer set, and/or artificially increase the extinction coefficient of some of the layers to represent the reduced likelihood of light reaching those layers.

In some implementations, calculation of the first spectrum can include calculating a stack reference R_(STACK1)

$R_{{STACK}\; 1} = \frac{E_{P} - \frac{H_{P}}{\mu_{P}}}{E_{P} + \frac{H_{P}}{\mu_{P}}}$

where for each layer j>0, E_(j) and H_(j) are calculated as

$\begin{bmatrix} E_{j} \\ H_{j} \end{bmatrix} = {\begin{bmatrix} {\cos \; g_{j}} & {\frac{}{u_{j}}\sin \; g_{j}} \\ {{\mu}_{j}\sin \; g_{j}} & {\cos \; g_{j}} \end{bmatrix}\begin{bmatrix} E_{j - 1} \\ H_{j - 1} \end{bmatrix}}$

where E₀ is 1 and H₀ is μ₀, and where for each layer j≧0, μ_(j)=(n_(j)−ik_(j))·cos φ_(j) and g_(j)=2π(n_(j)−ik_(j))·t_(j)·cos φ_(j)/λ, where n_(j) is the index of refraction of layer j, k_(j) is an extinction coefficient of layer j, t_(j) is the thickness of layer j, φ_(j) is the incidence angle of the light to layer j, and λ is the wavelength.

Similarly, the second spectrum can be calculated including a stack reflectance R_(STACK2)

$R_{{STACK}\; 2} = \frac{E_{P} - \frac{H_{P}}{\mu_{P}}}{E_{P} + \frac{H_{P}}{\mu_{P}}}$

where for each layer j>0, E_(j) and H_(j) are calculated as

$\begin{bmatrix} E_{j} \\ H_{j} \end{bmatrix} = {\begin{bmatrix} {\cos \; g_{j}} & {\frac{}{u_{j}}\sin \; g_{j}} \\ {{\mu}_{j}\sin \; g_{j}} & {\cos \; g_{j}} \end{bmatrix}\begin{bmatrix} E_{j - 1} \\ H_{j - 1} \end{bmatrix}}$

where E₀ is 1 and H₀ is μ₀, and where for each layer j≧0, μ_(j)=(n_(j)−i(k_(j)+m_(j)))·cos φ_(j) and g_(j)=2π(n_(j)−i(k_(j)+m_(j)))·t_(j) cos φ_(j)/λ, where n_(j) is an index of refraction of layer j, k_(j) is an extinction coefficient of layer j, m_(j) is the amount to increase the extinction coefficient of layer j,·t_(j) is the thickness of layer j, φ_(j) is the incidence angle of the light to layer j, and λ is the wavelength.

The calculation of spectrum R_(LIBRARY) can be iterated over multiple values for X and Y. For example, X can vary between 0.0 and 1.0 at 0.1 intervals and Y can vary between 0.0 and 1.0 at 0.1 intervals. A potential advantage of this technique is generation of reference spectra that can correspond to different concentrations of metal in the measured spot on the substrate, thus improving likelihood of finding a good matching reference spectra and improving accuracy and reliability of the optical monitoring system.

The controller software may receive user input identifying a plurality of different metal contribution percentages for the metal layer, which may include receiving user input identifying a first number of different contribution percentages for the first stack and receiving user input identifying a second number of different contribution percentages for the second stack. The plurality of different metal contribution percentages can be calculated from the first number of different contribution percentages and the second number of different contribution percentages.

In some implementations, the optical model can at least partially accounts for diffraction effects generated by a repeating feature on the substrate. In this case, at least one of the input parameters represents a characteristic of the repeating feature. The diffraction effects can be calculated using rigorous coupled waveform analysis. In particular, rigorous coupled waveform analysis (RCWA) can be used to model and calculate the diffraction effects. RCWA equations can be used to generate a reflectance R for each wavelength, and then to determine a diffraction efficiency at each wavelength.

Details of RCWA are laid out “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings” by Moharam et. al, and “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach” by Moharam et. al., each of which is incorporated by reference. For example, for optically modeling of a “1-D” diffraction grating, equations 24-26 from “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach” can be used to generate R for each wavelength, and the diffraction efficiency can be determined at each wavelength via equations 25 and 45 from “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings.” There are alternative techniques, e.g., described in “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity” by Lifeng Li. But in these various techniques, the model includes diffraction caused by the repeating structure.

Certain parameters can vary, e.g., due to process variations. As an example, process variations can happen with a substrate, among substrates of a single cassette, or from lot to lot. As noted above, for any parameters that are expected to vary, a fab facility or the manufacture of the equipment can set some ranges that also include incremental values or number of increments. Based on an optical model and by spanning the ranges of the parameters that can vary, the controller software generates a number of reference spectra. The generated reference spectra define the expected spectra for the substrate being polished. The in-situ or in-sequence optical monitoring system of a polishing apparatus can measure a spectrum of the substrate and find a best matching reference spectrum for the measured spectrum.

A problem is that as the number of input parameters that can vary grows, and as the number of increments e.g., ten or more for each parameter increases to e.g., 100 or more, the number of reference spectra in the library increases rapidly. For example, with six input parameters (e.g., thickness, index of refraction and extinction coefficient for each of two layers) and just ten possible values per input parameter, the library can reach one million reference spectra, which may be impractical for searching. Moreover, in some examples, the monitoring system samples an area of a substrate that covers more than one stack of layers causing the spectral contribution of different stack layers. A number of different reference spectra are calculated by combining the spectra of the different stack layers with all possible coverage percentages, further increasing the size of the reference spectra. Generating the reference spectra in memory and finding a reliable real-time match can strain the memory and/or the processing power if the size of the reference spectra is large.

The size of the library is reduced if one of the varying parameters is replaced with a fixed value. For example a parameter value for a particular substrate can be determined from an upstream measurement of the particular substrate before polishing occurs, e.g., the parameter value can be measured at a stand-alone metrology station or at an in-sequence monitoring system (e.g., the system described above with respect to FIG. 2) or before measurement at an in-sequence monitoring station occurs, e.g., the parameter value can be measured at a stand-alone metrology station. This parameter value is fed via a fab host to the controller 190. As an example the controller software can receive one or more values of one or more of parameters, e.g., the thickness, index of refraction, or extinction coefficient of one or more of the layers. To generate the reference spectra, the controller 190 holds the one or more parameters at the known values, and iterates on the remaining parameters.

This technique can also be applied to a gradient-descent type of search. Instead of producing one large library, gradient descent search will iterate through the varying parameters in real time to find the optimal solution. By feeding forward one of the parameters, gradient descent search has one less parameter to search through.

In some implementations, the substrate includes an underlying first layer and a second layer on top of the first layer. Before polishing, an in-line or stand-alone metrology system measures the substrate and determines the indices of refraction and extinction coefficients of both layers. The substrate moves to an in-sequence metrology station of a polishing station. The controller receives the indices of refraction and extinction coefficients of both layers as measured values. The metrology system also measures the thickness of both layers. Because of possible process variations, a range of thickness and an incremental value is assigned to the thickness of the underlying layer. Also, a range of thickness and an incremental value is assigned to the top layer that is being polished. The controller software generates reference spectra based on the thicknesses that can vary and the indices of refractions and extinction coefficients that are known and fixed. The reference spectra are used in an in-sequence metrology system of a polishing apparatus and the metrology system finds a best matching reference spectrum for the measured spectrum and based on the best matched reference spectrum determines the varying parameters, e.g., the thicknesses.

In some implementations, the substrate includes an underlying layer and an overlying layer on top of the underlying layer. Before polishing, an in-sequence metrology system measures the substrate and finds the indices of refraction and extinction coefficients of both layers. The substrate moves to polishing station having an in-situ monitoring system. The polishing station receives the indices of refraction and extinction coefficients of both layers as known values. The metrology system also measures the thickness of both layers. Because of possible process variations, a range of thickness and an incremental value is assigned to the thickness of the underlying layer. Also, a range of thickness and an incremental value is assigned to the top layer that is being polished. The controller software generates reference spectra based on the thicknesses that can vary and the indices of refractions and extinction coefficients that are known and fixed. The reference spectra are used in an in-situ metrology system of a polishing apparatus and the metrology system finds a best matching reference spectrum for the measured spectrum and based on the best matched reference spectrum determines the varying parameters, e.g., the thicknesses.

Referring to FIGS. 4 and 5, a measured spectrum 300 (see FIG. 4) can be compared to reference spectra 320 from one or more libraries 310 (see FIG. 5). As used herein, a library of reference spectra is a collection of reference spectra which represent substrates that share a property in common. However, the property shared in common in a single library may vary across multiple libraries of reference spectra. For example, two different libraries can include reference spectra that represent substrates with two different underlying thicknesses. For a given library of reference spectra, variations in the upper layer thickness, rather than other factors (such as differences in wafer pattern, underlying layer thickness, or layer composition), can be primarily responsible for the differences in the spectral intensities.

In addition to reference spectra that can be calculated from theory, e.g., an optical model can be used to calculate a reference spectrum for a given outer layer thickness D. A value representing the time in the polishing process at which the reference spectrum would be collected can be calculated, e.g., by assuming that the outer layer is removed at a uniform polishing rate. For example, the time Ts for a particular reference spectrum can be calculated simply by assuming a starting thickness D0 and uniform polishing rate R (Ts=(D0−D)/R). As another example, linear interpolation between measurement times T1, T2 for the pre-polish and post-polish thicknesses D1, D2 (or other thicknesses measured at the metrology station) based on the thickness D used for the optical model can be performed (Ts=T2−T1*(D1−D)/(D1−D2)).

Reference spectra for different libraries can be calculated from theory, e.g., spectra for a first library can be calculated using the optical model with the underlying layer having a first thickness, and spectra for a second library can be calculated using the optical model with the underlying layer having a different one thickness. For example, this disclosure uses a copper substrate for generating the library and later for spectra measurements.

In some implementations, each reference spectrum 320 (see FIG. 5) is assigned an index value 330. In general, each library 310 can include many reference spectra 320, e.g., one or more, e.g., exactly one, reference spectra for each calculated platen rotation over the expected polishing time of the substrate. This index 330 can be the value, e.g., a number, representing the time in the polishing process at which the reference spectrum 320 is expected to be observed. The spectra can be indexed so that each spectrum in a particular library has a unique index value. The indexing can be implemented so that the index values are sequenced in an order in which the spectra of a test substrate were measured. An index value can be selected to change monotonically, e.g., increase or decrease, as polishing progresses. In particular, the index values of the reference spectra can be selected so that they form a linear function of time or number of platen rotations (assuming that the polishing rate follows that of the model or test substrate used to generate the reference spectra in the library). For example, the index value can be proportional, e.g., equal, to an estimated number of platen rotations at which the reference spectra was measured for the test substrate or would appear in the optical model. Thus, each index value can be a whole number. The index number can represent the expected platen rotation at which the associated spectrum would appear.

The reference spectra and their associated index values can be stored in a reference library. For example, each reference spectrum 320 and its associated index value 330 can be stored in a record 340 of database 350. The database 350 of reference libraries of reference spectra can be implemented in memory of the computing device of the polishing apparatus. In some implementation, one or more of the reference spectra, e.g., all of the reference spectra are calculated on the fly.

As noted above, for each zone of each substrate, based on the sequence of measured spectra or that zone and substrate, the controller 190 can be programmed to generate a sequence of best matching spectra. A best matching reference spectrum can be determined by comparing a measured spectrum to the reference spectra from a particular library.

In some implementations, the best matching reference spectrum can be determined by calculating, for each reference spectrum, a sum of squared differences between the measured spectrum and the reference spectrum. The reference spectrum with the lowest sum of squared differences has the best fit. Other techniques for finding a best matching reference spectrum are possible, e.g., lowest sum of absolute differences.

In some implementations, the best matching reference spectrum can be determined by using a matching technique other than sum of squared differences. In one implementation, for each reference spectrum, a cross-correlation between the measured spectrum and the reference spectrum is calculated, and the reference spectrum with the greatest correlation is selected as the matching reference spectrum. A potential advantage of cross-correlation is that it is less sensitive to lateral shift of a spectrum, and thus can be less sensitive to underlying thickness variation. In order to perform the cross-correlation, the leading and trailing ends of the measured spectrum can be padded with “zeros” to provide data to compare against the reference spectrum as the reference spectrum is shifted relative to the measured spectrum. Alternatively, the leading end of the measured spectrum can be padded with values equal to the value at the leading edge of the measured spectrum, and he trailing end of the measured spectrum can be padded with values equal to the value at the trailing edge of the measured spectrum. Fast Fourier transforms can be used to increase the speed of calculation of the cross-correlation for real-time application of the matching technique.

The fitting of an optical model to the measured spectrum can also be performed using the measured value as a seed value in the fitting algorithm. That is, rather than generate reference spectra, the measured parameter values are used as seed values in the fitting algorithm.

Once a seed value is identified, regression can be used to fit the optical model to the measured spectrum. Examples of regression techniques include Levenberg-Marquardt (L-M)—which utilizes a combination of Gradient Descent and Gauss-Newton; Fminunc( )—a matlab function; lsqnonlin( )—matlab function that uses the L-M algorithm; and simulated annealing. In addition, non-regression techniques, such as the simplex method, can be used to optimize the parameters.

Now referring to FIG. 6, which illustrates the results for only a single zone of a single substrate, the index value of each of the best matching spectra in the sequence can be determined to generate a time-varying sequence of index values 212. This sequence of index values can be termed an index trace 210. In some implementations, an index trace is generated by comparing each measured spectrum to the reference spectra from exactly one library. In general, the index trace 210 can include one, e.g., exactly one, index value per sweep of the optical monitoring system below the substrate.

In summary, each index trace includes a sequence 210 of index values 212, with each particular index value 212 of the sequence being generated by selecting the index of the reference spectrum from a given library that is the closest fit to the measured spectrum. The time value for each index of the index trace 210 can be the same as the time at which the measured spectrum was measured.

A monitoring technique is used to detect clearing of the second layer and exposure of the underlying layer or layer structure. For example, exposure of the first layer at a time TC can be detected by a sudden change in the motor torque or total intensity of light reflected from the substrate, or from dispersion of the collected spectra as discussed in greater detail below.

As shown in FIG. 7, a function, e.g., a polynomial function of known order, e.g., a first-order function (e.g., a line 214) is fit to the sequence of index values of spectra collected after time TC, e.g., using robust line fitting. Index values for spectra collected before the time TC are ignored when fitting the function to the sequence of index values. Other functions can be used, e.g., polynomial functions of second-order, but a line provides ease of computation. Polishing can be halted at an endpoint time TE that the line 214 crosses a target index IT.

FIG. 8 shows a flow graph of a method 800 for controlling a polishing operation of a product substrate. The controller stores an optical model for a layer stack (step 810). The model includes a number of input parameters including a first and a second parameter. Examples of the input parameters include one or more of thicknesses, indices of refraction, or extinction coefficients of each of the layers in the layer stack. The controller stores a number of default values for the first parameter (step 820). The default values can be generated by defining a range of variation for the first parameter and additionally defining an incremental value or a number of increments across the range. An optical property of a substrate is measured and a second value for the second parameter is generated (step 830). The measurement is performed in a stand-alone or in-sequence metrology station. The controller calculates a number of reference spectra using the optical model while using the value of the second parameter and each one of the default values of the first parameter (step 840). The substrate is transferred from the stand-alone or in-sequence metrology station to an in-sequence or in-situ monitoring system of a chemical polishing apparatus. The spectrum of the substrate is measured by the in-sequence or in-situ monitoring system (step 850). The controller determines a best matching reference spectrum from the reference spectra for the measured spectrum and based on the matched reference spectrum determines the first value (step 860). The substrate is polished by the polishing apparatus and a polishing endpoint or a polishing parameter is adjusted (step 870) where the adjustment is based on the first value.

In some implementations, the first parameter is a thickness of the outermost layer of the substrate and the second parameter is an index of refraction or extinction coefficient of the outermost layer or a thickness of an underlying layer of the substrate.

In some implementations, the optical property of the substrate is measured at the stand-alone metrology station to generate the second value, and the spectrum is measured with the in-sequence monitoring system to provide the measured spectrum. Alternatively, the optical property of the substrate is measured at the in-sequence metrology station to generate the second value, and the spectrum is measured with the in-situ monitoring system to provide the measured spectrum.

In some implementations, a plurality of second values are calculated based on the second value, a default range, and a default increment and a reference spectrum is calculated using the optical model for each combination of a first value from the plurality of first values and a second value from the plurality of second values.

FIG. 9 shows a flow graph of a method 900 for controlling a polishing operation. The controller stores an optical model for a layer stack (step 910). The model includes a number of parameters including a first and a second parameter. The parameters are one or more of thicknesses, indices of refraction, or extinction coefficients of the layer stack. An optical property of a substrate is measured and a second value for the second parameter is generated (step 920). The measurement is performed in a stand-alone or in-sequence metrology station. The substrate is transferred from the stand-alone or in-sequence metrology station to an in-sequence or in-situ monitoring system of a chemical polishing apparatus. The spectrum of the substrate is measured by the in-sequence or in-situ monitoring system (step 930). The controller fits the optical model to the measured spectrum and finds a first value for the first parameter (step 940). The fitting includes finding a first value of the first parameter that provides a minimum difference between an output spectrum of the optical model and the measured spectrum. The substrate is polished by the polishing apparatus and a polishing endpoint or a polishing parameter is adjusted (step 950) where the adjustment is based on the first value associated with the fitted optical model.

In some implementations, the fitting includes holding the second parameter at the second value or using the second value of the second parameter as a seed value in searching for the minimum difference.

In some implementations, the first parameter is a thickness of the outermost layer of the substrate and the second parameter is an index of refraction or extinction coefficient of the outermost layer or a thickness of an underlying layer of the substrate.

In some implementations, the optical property of the substrate is measured at the stand-alone metrology station to generate the second value, and the spectrum is measured with the in-sequence monitoring system to provide the measured spectrum. Alternatively, the optical property of the substrate is measured at the in-sequence metrology station to generate the second value, and the spectrum is measured with the in-situ monitoring system to provide the measured spectrum.

Embodiments of the invention and all of the functional operations described in this specification can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structural means disclosed in this specification and structural equivalents thereof, or in combinations of them. Embodiments of the invention can be implemented as one or more computer program products, i.e., one or more computer programs tangibly embodied in a machine readable storage media, for execution by, or to control the operation of, data processing apparatus, e.g., a programmable processor, a computer, or multiple processors or computers. A computer program (also known as a program, software, software application, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file. A program can be stored in a portion of a file that holds other programs or data, in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers at one site or distributed across multiple sites and interconnected by a communication network.

The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).

The above described polishing apparatus and methods can be applied in a variety of polishing systems. Either the polishing pad, or the carrier heads, or both can move to provide relative motion between the polishing surface and the substrate. For example, the platen may orbit rather than rotate. The polishing pad can be a circular (or some other shape) pad secured to the platen. Some aspects of the endpoint detection system may be applicable to linear polishing systems, e.g., where the polishing pad is a continuous or a reel-to-reel belt that moves linearly. The polishing layer can be a standard (for example, polyurethane with or without fillers) polishing material, a soft material, or a fixed-abrasive material. Terms of relative positioning are used; it should be understood that the polishing surface and substrate can be held in a vertical orientation or some other orientation.

Particular embodiments of the invention have been described. Other embodiments are within the scope of the following claims. 

What is claimed is:
 1. A method of controlling a polishing operation, comprising: storing an optical model for a layer stack having a plurality of layers, the optical model having a plurality of input parameters including a first parameter and a second parameter; storing data defining a plurality of default first values for the first parameter; measuring an optical property of a substrate at a stand-alone or in-sequence metrology station to generate a second value for the second parameter; for each first value from the plurality of first values, calculating a reference spectrum using the optical model based on the first value and the second value, to generate a plurality of reference spectra; transporting the substrate from the stand-alone or in-sequence metrology station to an in-sequence or in-situ monitoring system of a chemical mechanical polishing apparatus; measuring a spectrum with the in-sequence or in-situ monitoring system to provide a measured spectrum; determining a best matching reference spectrum from the plurality of reference spectra that provides a best match to the measured spectrum; determining the first value associated with the best matching reference spectrum; polishing the substrate with the polishing apparatus; and adjusting a polishing endpoint or a polishing parameter of the polishing apparatus based on the first value associated with the best matching reference spectrum.
 2. The method of claim 1, wherein the first parameter comprises a thickness of an outermost layer of the substrate.
 3. The method of claim 2, wherein the second parameter comprises an index of refraction or extinction coefficient of the outermost layer or a thickness of an underlying layer of the substrate.
 4. The method of claim 1, comprising measuring the optical property of the substrate at the stand-alone metrology station to generate the second value, and measuring the spectrum with the in-sequence monitoring system to provide the measured spectrum.
 5. The method of claim 1, comprising measuring the optical property of the substrate at the in-sequence metrology station to generate the second value, and measuring the spectrum with the in-situ monitoring system to provide the measured spectrum.
 6. The method of claim 1, comprising calculating a plurality of second values based on the second value, a default range, and a default increment.
 7. The method of claim 6, comprising calculating a reference spectrum using the optical model for each combination of a first value from the plurality of first values and a second value from the plurality of second values.
 8. A method of controlling a polishing operation, comprising: storing an optical model for a layer stack having a plurality of layers, the optical model having a plurality of input parameters including a first parameter and a second parameter; measuring an optical property of a substrate at a stand-alone or in-sequence metrology station to generate a second value for a second parameter of the plurality of optical parameters; transporting the substrate from the stand-alone or in-sequence metrology station to an in-sequence or in-situ monitoring system of a chemical mechanical polishing apparatus; measuring a spectrum with the in-sequence or in-situ monitoring system to provide a measured spectrum; fitting the optical model to the measured spectrum, the fitting including finding a first value of the first parameter that provides a minimum difference between an output spectrum of the optical model and the measured spectrum, the fitting including holding the second parameter at the second value or using the second value of the second parameter as a seed value in searching for the minimum difference; polishing the substrate with the polishing apparatus; and adjusting a polishing endpoint or a polishing parameter of the polishing apparatus based on the first value associated with the fitted optical model.
 9. The method of claim 8, wherein the first parameter comprises a thickness of an outermost layer of the substrate.
 10. The method of claim 9, wherein the second parameter comprises an index of refraction or extinction coefficient of the outermost layer or a thickness of an underlying layer of the substrate.
 11. The method of claim 8, comprising measuring the optical property of the substrate at the stand-alone metrology station to generate the second value, and measuring the spectrum with the in-sequence monitoring system to provide the measured spectrum.
 12. The method of claim 8, comprising measuring the optical property of the substrate at the in-sequence metrology station to generate the second value, and measuring the spectrum with the in-situ monitoring system to provide the measured spectrum.
 13. The method of claim 8, wherein the fitting includes holding the second parameter at the second value.
 14. The method of claim 8, wherein the fitting includes using the second value of the second parameter as a seed value in searching for the minimum difference. 